Related papers: Applications of the abc conjecture to powerful num…

Let $c$ be a positive odd integer and $R$ a set of $n$ primes coprime with $c$. We consider equations $X + Y = c^z$ in three integer unknowns $X$, $Y$, $z$, where $z > 0$, $Y > X > 0$, and the primes dividing $XY$ are precisely those in…

The ternary Goldbach conjecture (or three-prime conjecture) states that every odd number greater than 5 can be written as the sum of three primes. The purpose of this book is to give the first proof of the conjecture, in full.

The ternary Goldbach conjecture, or three-primes problem, states that every odd number $n$ greater than $5$ can be written as the sum of three primes. The conjecture, posed in 1742, remained unsolved until now, in spite of great progress in…

Let $a$ and $m>0$ be integers. We show that for any integer $b$ relatively prime to $m$, the set $\{a^n+bn:\ n=1,\ldots,m^2\}$ contains a complete system of residues modulo $m$. We also pose several conjectures for further research; for…

We consider $n$-sided dice whose face values lie between $1$ and $n$ and whose faces sum to $n(n+1)/2$. For two dice $A$ and $B$, define $A \succ B$ if it is more likely for $A$ to show a higher face than $B$. Suppose $k$ such dice…

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…

For given integers a,b, and j at least 1 we determine the set of integers n for which a^n-b^n is divisible by n^j. For j=1,2, this set is usually infinite; we find explicitly the exceptional cases for which a,b the set is finite. For j=2,…

Problem 2 at the 56th International Mathematical Olympiad (2015) asks for all triples (a,b,c) of positive integers for which ab-c, bc-a, and ca-b are all powers of 2. We show that this problem requires only a primitive form of arithmetic,…

In this paper we consider integers in base 10 like $abc$, where $a$, $b$, $c$ are digits of the integer, such that $abc^2 - (abc \cdot cba) \; = \; \pm n^2$, where $n$ is a positive integer, as well as equations $abc^2 - (abc \cdot cba) \;…

This document seeks to prove there are infinitely many primes whose difference is 2, referred to as twin prime pairs. This proof's methodology involves constructing a function that approximates the number of positive integers, less than a…

We discuss conjectures related to the following two conjectures: (1) for each complex numbers x_1,...,x_n there exist rationals y_1,...,y_n \in [-2^{n-1},2^{n-1}] such that \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

In this paper, we will show that the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers is unusually large. This generalizes the very recent results by the author and by A. Dubickas, which are both…

Let $f$ be a positive multiplicative function and let $k\geq 2$ be an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\rightarrow +\infty$, in the sense that $\sum_{p}|f(p)-1|=\infty$, there exists…

We examine Euclid's lemma that if $p$ is a prime number such that $p | ab$, then $p$ divides at least one of $a$ or $b$. Specifically, we consider the common misapplication of this lemma to numbers that are not prime, as is often made by…

By Zeckendorf's Theorem, every positive integer is uniquely written as a sum of non-adjacent terms of the Fibonacci sequence, and its converse states that if a sequence in the positive integers has this property, it must be the Fibonacci…

This short article is aimed at educators and teachers of mathematics.Its goal is simple and direct:to explore some of the basic/elementary properties of proper rational numbers.A proper rational number is a rational which is not an integer.…

The Diophantine equation 4/n=1/x+1/y+1/z for a Pythagorean prime n is split into two independent Diophantine equations, which correspond to two different types of solution. The solvability of these equations forces certain restrictions on…

For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…

A conjecture of Mordell states that if $p$ is a prime and $p$ is congruent to $3$ mod $4$, then $p$ does not divide $y$ where $(x,y)$ is the fundamental solution to $x^{2}-py^{2}=1$. The conjecture has been verified for primes not exceeding…